Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

Abstract

We present a comprehensive Monte Carlo study of two-dimensional bond percolation with algebraically decaying connection probabilities $p(r)\propto 1/r^{2+σ}$, establishing the universality diagram in the long-range (LR) regime for $σ\le2$. Using the event-based ensemble method, we simulate systems with linear sizes up to $L=16384$ and investigate three universality regimes: LR Wilson--Fisher (WF) A ($1<σ\le2$), LR Wilson--Fisher B ($2/3<σ\le1$), and LR mean-field (MF) ($0<σ\le2/3$). In the LR-WF-B regime, the anomalous dimension is consistent with $η=2-σ$, in agreement with mathematical results for $2/3<σ<1$, while the correlation-length exponent $ν(σ)$ exhibits nontrivial, non-Gaussian variation. In the LR-WF-A regime, although $η$ remains close to $2-σ$ for smaller $σ$, statistically resolvable deviations $δη(σ)=η-(2-σ)>0$ start to appear near $σ\simeq3/2$ and grow toward the short-range crossover at $σ=2$. Finally, by complementing the event-based simulations with conventional ensemble simulations, we reveal the coexistence of complete-graph asymptotics and LR Gaussian-fixed-point scaling in the LR-MF regime. These results further clarify the critical properties in long-range percolation and provide crucial benchmarks for long-range statistical systems.